Unit 1 – Chapter 5.

Unit 1 – Chapter 5

Unit 1 Section 5.1 – Write Linear Equations in Slope-Intercept Form
Chapter 5 Review

Warm-Up – 5.1

Lesson 5.1, For use with pages 282-291
Find the slope of the line that passes through the points. 1. (2, –1), (4, 0) 1 2 ANSWER 2. (–1, –3), (1, 5) ANSWER 4

3. A landscape architect charges $75 for a consulting fee and $30 per hour. Write an equation that shows the cost C as a function of time t (in hours). ANSWER C = 30t + 75

Evaluate 1. f(x) = - x – 2, x = 2 ANSWER F(2) = -(2) -2 = -4 2. f(x) = 2x-3. Find x if f(x) = 11 ANSWER x = 7 because f(7) = 2(7) – 3 = 11

Vocabulary – 5.1 Y-intercept Where graph crosses the y-axis
Where the story “starts” Slope How fast something changes! AKA Unit rate, steepness, rate of change, constant of variation, etc. Slope-Intercept Form Y=mx+b Standard Form Ax + By = C

Notes – 5.1 – Write LE in Slope-Int Form.
Slope – Intercept form Y = Mx + b Need TWO things to write an equation in Slope-Int form. There are many ways to get them! Slope – rise/run, y2-y1/x2-x1, how much over how long, draw triangles on graphs, etc. Y - Intercept – Read graph, set x=0 and solve for y, find “b” in slope-intercept form

Examples 5.1

Use slope and y-intercept to write an equation
EXAMPLE 1 Use slope and y-intercept to write an equation Write an equation of the line with a slope of – 2 and a y -intercept of 5. y = mx + b Write slope-intercept form. y = – 2x + 5 Substitute – 2 for m and = for b.

Standardized Test Practice
EXAMPLE 2 Standardized Test Practice Which equation represents the line shown? A y = – x + 3 2 5 B C y = – x + 1 D y = 3x + = – The slope of the line is rise run –2 5 2 . The line crosses the y-axis at (0, 3). So, the y-intercept is 3. y = mx + b Write slope-intercept form. 2 y = – x + 3 5 2 Substitute – for m and 3 for b. 5

EXAMPLE 2 Standardized Test Practice ANSWER The correct answer is A. B D C A

Write an equation of the line with the given slop and y-intercept.
GUIDED PRACTICE for Examples 1 and 2 Write an equation of the line with the given slop and y-intercept. 1. Slope is 8; y-intercept is –7. SOLUTION y = mx + b Write slope-intercept form. y = 8x –7 Substitute 8 for m and –7 for b.

2. Slope is ; y intercept is – 3. 3 4
GUIDED PRACTICE for Examples 1 and 2 2. Slope is ; y intercept is – 3. 3 4 SOLUTION y = m x + b Write slope-intercept form. y = 3 4 x – 3 Substitute for m and – 3 for b. 3 4

EXAMPLE 3 Write an equation of a line given two points Write an equation of the line shown.

Write an equation of a line given two points
EXAMPLE 3 Write an equation of a line given two points SOLUTION STEP 1 Calculate the slope. x2 – x1 3 3 – 0 y2 – y1 = m – 1 – (– 5) 4 STEP 2 Write an equation of the line. The line crosses the y-axis at (0, – 5). So, the y-intercept is – 5. y = mx + b Write slope-intercept form. y = x – 5 4 3 Substitute for m and 5 for b. 4 3 –

EXAMPLE 4 Write a linear function Write an equation for the linear function f with the values f(0) = 5 and f(4) = 17. SOLUTION STEP 1 Write f(0) = 5 as (0, 5) and f (4) = 17 as (4, 17). STEP 2 Calculate the slope of the line that passes through (0, 5) and (4, 17). x2 – x1 4 – 0 y2 – y1 = m 17 – 5 4 12 3

Write a linear function
EXAMPLE 4 Write a linear function STEP 3 Write an equation of the line. The line crosses the y-axis at (0, 5). So, the y-intercept is 5. y = mx + b Write slope-intercept form. y = 3x + 5 Substitute 3 for m and 5 for b. ANSWER The function is f(x) = 3x + 5.

3. Write an equation of the line shown.
GUIDED PRACTICE for Examples 3 and 4 3. Write an equation of the line shown. SOLUTION STEP 1 Calculate the slope. = 4 – 0 m x2 – x1 y2 – y1 – 1 – 1 2 –1 STEP 2 Write an equation of the line. The line comes the y-axis at (0, 1). So, the y-intercept is 1. y = mx + b Write slope-intercept form. 2 –1 x + 1 y = – Substitute for m and 1 for b. 2 –1

GUIDED PRACTICE for Examples 3 and 4 Write an equation for the linear function f with the given values. 4. f(0) = –2, f(8) = 4 SOLUTION STEP 1 Write f(0) = –2 as (0, –2 ) and f (8) = 4 as (8, 4). STEP 2 Calculate the slope of the line that passes through (0, –2) and (8, 4). = m 8 6 4 3 4 – (– 2) 8 – 0

GUIDED PRACTICE for Examples 3 and 4 STEP 3
Write an equation of the line. The line comes the y-axis at (0, – 2). So, the y-intercept is – 2. y = mx + b Write slope-intercept form. y = x – 2 3 4 The function is f (x) = 3 4 x – 2 ANSWER

Warm-Up – 5.2

Find the equation of the line that passes through the points. 1. (0, –2), (1, 3) ANSWER Y = 5x -2 2. (3, 2), (0, –2) ANSWER Y = 4/3 x - 2 23

3. Bill wants a video game that costs $75. After borrowing $75 from his parents, he is paying back $5 per week. Write an equation that tells how much money Bill owes after x weeks. ANSWER y = –5x + 75 Using the slope-intercept form, find b if slope = 2, x = 2 and y = 9. Then find the equation of the line. ANSWER b = 5 Equation : y = 2x + 5 24

Vocabulary – 5.2 Slope-Intercept Form Y = mx + b Y-intercept
Where a graph crosses the y –axis X = 0 X-intercept Where a graph crosses the x axis Y=0

Notes – 5.2 – Use Linear Eqns in Slope-Int. Form.
To use slope intercept form, I need two pieces of information: Slope Y-intercept If you have ONE point and the slope, you can find the y-intercept by “doing the dance!” Plug in the slope and (x,y) to the slope-int form and solve for b. To find out if 3 points are on the same line, Find the lin. Eqn. for the first two points. Plug in the (x,y) coordinates from the third point to the eqn. and see if it works.

Examples 5.2

Write an equation given the slope and a point
EXAMPLE 1 Write an equation given the slope and a point Write an equation of the line that passes through the point (– 1, 3) and has a slope of – 4. SOLUTION STEP 1 Identify the slope. The slope is – 4. STEP 2 Find the y-intercept. Substitute the slope and the coordinates of the given point in y = m x + b. Solve for b. y = m x + b Write slope-intercept form. 3 = – 4(– 1) + b Substitute – 4 for m, – 1 for x, and 3 for y.

Write an equation given the slope and a point
EXAMPLE 1 Write an equation given the slope and a point – 1 = b Solve for b. STEP 3 Write an equation of the line. y = m x + b Write slope-intercept form. y = – 4x – 1 Substitute – 4 for m and – 1 for b.

Identify the slope. The slope is 2.
GUIDED PRACTICE for Example 1 Write an equation of the line that passes through the point (6, 3) and has a slope of 2. SOLUTION STEP 1 Identify the slope. The slope is 2. STEP 2 Find the y-intercept. Substitute the slope and the coordinates of the given point in y = m x + b. Solve for b. y = m x + b Write slope-intercept form. 3 = 2(6) + b Substitute 3 for y, 2 for m and 6 for x.

Write an equation of the line.
GUIDED PRACTICE for Example 1 – 9 = b Solve for b. STEP 3 Write an equation of the line. y = m x + b Write slope-intercept form. y = 2x – 9 Substitute 2 for m and – 9 for b.

Write an equation given two points
EXAMPLE 2 Write an equation given two points Write an equation of the line that passes through (– 2, 5) and (2, – 1). SOLUTION STEP 1 Calculate the slope. 3 m = y2 – y1 x2 – x1 = – 1 – 5 2 – (– 2) – 6 4 – 2 STEP 2 Find the y-intercept. Use the slope and the point (– 2, 5). y = m x + b Write slope-intercept form.

Write an equation given two points
EXAMPLE 2 Write an equation given two points 5 = – 3 2 (– 2) – b Substitute – for m, – 2 for x, and 5 for y. 3 2 2 = b Solve for b. STEP 3 Write an equation of the line. y = m x + b Write slope-intercept form. y = – 3 2 x + 2 Substitute – 3 2 for m and 2 for b.

Find the y-intercept. Use the slope and the point (1, –2).
GUIDED PRACTICE for Examples 2 and 3 2. Write an equation of the line that passes through (1, –2) and (–5, 4). SOLUTION STEP 1 Calculate the slope. m = y2 – y1 x2 – x1 = 4 –(– 2) –5 – 1 6 –6 – 1 STEP 2 Find the y-intercept. Use the slope and the point (1, –2). y = m x + b Write slope-intercept form.

GUIDED PRACTICE for Examples 2 and 3 –2 = –1 (1) + b Substitute – 2 for y and – 1 for m. – 1 = b Solve for b. STEP 3 Write an equation of the line. y = m x + b Write slope-intercept form. y = – x – 1 Substitute – 1 for m and – 1 for b.

Find the y-intercept. Use the slope and the point (-4, –2).
EXAMPLE 3 2. Do these three points lie on the same line? (-4,-2), (2,2.5) and (8,7). SOLUTION STEP 1 Calculate the slope. m = y2 – y1 x2 – x1 = 2.5–(- 2) 2 – (-4) 4.5 6 = 3/4 STEP 2 Find the y-intercept. Use the slope and the point (-4, –2). y = m x + b Write slope-intercept form.

GUIDED PRACTICE for Examples 2 and 3 –2 = 3/4 (-4) + b Substitute – 2 for y and – 4 for x and ¾ for m.. 1 = b Solve for b. STEP 3 Write an equation of the line. y = m x + b Write slope-intercept form. y = 3/4 x + 1 Substitute – 1 for m and – 1 for b. STEP 4 Plug in (8,7) to see if it’s a solution (7) = ¾ (8) + 1 7 = 6 + 1 7 = 7  so all three points are on the line

EXAMPLE 4 Solve a multi-step problem GYM MEMBERSHIP Your gym membership costs $33 per month after an initial membership fee. You paid a total of $228 after 6 months Write an equation that gives the total cost as a function of the length of your gym membership (in months). Find the total cost after 9 months. SOLUTION STEP 1 Identify the rate of change and starting value.

EXAMPLE 4 Solve a multi-step problem Rate of change, m: monthly cost, $33 per month Starting value, b: initial membership fee STEP 2 Write a verbal model. Then write an equation. C = t b 33 +

Solve a multi-step problem
EXAMPLE 4 Solve a multi-step problem STEP 3 Find the starting value. Membership for 6 months costs $228, so you can substitute 6 for t and 228 for C in the equation C = 33t + b. 228 = 33(6) + b Substitute 6 for t and 228 for C. 30 = b Solve for b. STEP 4 Write an equation. Use the function from Step 2.

EXAMPLE 4 Solve a multi-step problem STEP 4 Write an equation. Use the function from Step 2. C = 33t + 30 Substitute 30 for b. STEP 5 Evaluate the function when t = 9. C = 33(9) + 30 = 327 Substitute 9 for t. Simplify. Your total cost after 9 months is $327. ANSWER

EXAMPLE 5 Solve a multi-step problem BMX RACING In Bicycle Moto Cross (BMX) racing, racers purchase a one year membership to a track. They also pay an entry fee for each race at that track. One racer paid a total of $125 after 5 races. A second racer paid a total of $170 after 8 races. How much does the track membership cost? What is the entry fee per race?

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Identify the rate of change and starting value. Rate of change, m: entry fee per race Starting value, b: track membership cost STEP 2 Write a verbal model. Then write an equation. C = m r + b

EXAMPLE 5 Solve a multi-step problem STEP 3 Calculate the rate of change. This is the entry fee per race. Use the slope formula. Racer 1 is represented by (5, 125). Racer 2 is represented by (8, 170). 45 m = y2 – y1 x2 – x1 170 – 125 8 – 5 3 15 STEP 4 Find the track membership cost b. Use the data pair (5, 125) for racer 1 and the entry fee per race from Step 3.

EXAMPLE 5 Solve a multi-step problem C = mr + b Write the equation from Step 2. 125 = 15(5) + b Substitute 15 for m, 5 for r, and 125 for C. 50 = b Solve for b. ANSWER The track membership cost is $50. The entry fee per race is $15. Therefore C = 15x + 50.

Warm-Up – 5.3

Write an equation of the line. 1. passes through (3, 4), m = 3 ANSWER y = 3x – 5 2. passes through (–2, 2) and (1, 8) ANSWER y = 2x + 6 47

x ≤ −5 OR x ≥ 3 2|x+1| - 2 ≥ 6 Lesson 5.3, For use with pages 302-308
4. Solve and graph the following inequality 2|x+1| - 2 ≥ 6 x ≤ −5 OR x ≥ 3 ANSWER 3. Multiply both sides of the slope equation m = (y2-y1) (x2-x1) by (x2-x1). What do you get? ANSWER (y2-y1)= m (x2-x1) 48

Vocabulary – 5.3 – Point – slope form
Fairly rare way of writing a linear equation that includes the slope and coordinates of one point.

Notes – 5.3 – Lin. Eqns in point-slope form
Studied TWO ways to write Lin. Eqn’s Standard Form – Ax + By = C Slope-Intercept Form – Y = Mx + b The third way is the Point-Slope Form Looks like (y-y1) = m(x-x1) Need two things: Slope ONE point Can be converted to Standard Form or Slope-Intercept form. CAN look different for every point (which is why it’s not used very often!)

Examples 5.3

Write an equation in point-slope form
EXAMPLE 1 Write an equation in point-slope form Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2. Write point-slope form. y – y1 = m (x – x1) y + 3 = 2 (x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.

Write an equation in point-slope form for Example 1
GUIDED PRACTICE Write an equation in point-slope form for Example 1 Write an equation in point-slope form of the line that passes through the point (– 1, 4) and has a slope of – 2 . 1. y – y1 = m (x – x1) Write point-slope form. y – 4 = –2 (x +1) Substitute – 2 for m, 4 for y, and –1 for x.

Graph an equation in point-slope form
EXAMPLE 2 Graph an equation in point-slope form Graph the equation. y + 2 = (x – 3). 2 3

EXAMPLE 2 Graph an equation in point-slope form SOLUTION Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2). 2 3 Plot the point (3, – 2). Find a second point on the line using the slope. Draw a line through both points.

Graph an equation in point-slope form for Example 2
GUIDED PRACTICE Graph an equation in point-slope form for Example 2 Graph the equation. 2. y – 1 = (x – 2) – SOLUTION Because the equation is in point-slope form, you know that the line has a slope of –1 and passes through the point (2, 1). Plot the point (2, 1). Find a second point on the line using the slope. Draw a line through both points.

EXAMPLE 3 Use point-slope form to write an equation Write an equation in point-slope form of the line shown.

EXAMPLE 3 Use point-slope form to write an equation SOLUTION STEP 1 Find the slope of the line. = y1 – y2 x1 – x2 m 3 –1 – 1 –1 2 – 2 – 1

EXAMPLE 3 Use point-slope form to write an equation STEP 2 Write the equation in point-slope form. You can us either given point. Method 1 Method 2 Use (– 1, 3). Use (1, 1). y – y1 = m(x – x1) y – y1 = m(x – x1) y – 3 = – (x +1) y – 1 = – (x – 1) CHECK Check that the equations are equivalent by writing them in slope-intercept form. y – 3 = –x – 1 y – 1 = –x + 1 y = –x + 2 y = –x + 2

EXAMPLE 3 GUIDED PRACTICE Use point-slope form to write an equation for Example 3 Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). 3. STEP 1 Find the slope of the line. = y2 – y1 x2 – x4 m 4 –3 4 –2 1 2

EXAMPLE 3 GUIDED PRACTICE Use point-slope form to write an equation for Example 3 STEP 2 Write the equation in point-slope form. You can us either given point. Method 1 Method 2 Use (2, 3) Use (4, 4) y – y1 = m(x – x1) y – y1 = m(x – x1) y – 3 = (x – 2) 1 2 y – 4 = (x – 4) 1 2

Warm-Up – 5.4

Write an equation in point-slope form of the line that passes through the given points. 1. (1, 4), (6, –1) ANSWER y – 4 = –(x – 1) or y + 1 = –(x – 6) 2. ( –1, –2), (2, 7) ANSWER y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

3. Convert the equation y – 5 = 3(x – 3) to slope-intercept form AND Standard Form Slope Intercept y = 3x – 4 Standard Form -3x + y = -4 ANSWER Write the following equation on your whiteboard 2x + 3y = 6 and graph it on your calculator. HINT: What form does the equation have to be in for the calculator? Multiply both sides of the ORIGINAL equation by 2 and write that equation down on your whiteboard. Now graph the new equation on your calculator. What do you notice? New equation: 4x + 6y = 12 Graphs are identical! ANSWER

Vocabulary – 5.4 Standard Form of a Linear Equation Ax + By = C

Notes – 5.4 – Write Lin. Eqns in Standard Form
Standard form is GREAT for graphing intercepts Y– intercept  set x = 0 X –Intercept  set y = 0 Can convert point-slope form and slope intercept for to standard for by getting all the variables on one side and the constants on the other side. Equations that have a common multiple or factor are still equivalent.

Examples 5.4

EXAMPLE 1 Write equivalent equations in standard form Write two equations in standard form that are equivalent to 2x – 6y = 4. SOLUTION To write one equivalent equation, multiply each side by 2. To write another equivalent equation, multiply each side by 0.5. 4x – 12y = 8 x – 3y = 2

Write an equation from a graph
EXAMPLE 2 Write an equation from a graph Write an equation in standard form of the line shown. SOLUTION STEP 1 Calculate the slope. – 3 m = 1 – (–2) 1 –2 = 3 –1 STEP 2 Write an equation in point-slope form. Use (1, 1). y – y1 = m(x – x1) Write point-slope form. y – 1 = – 3(x – 1) Substitute 1 for y1, 23 for m and 1 for x1.

Write an equation from a graph
EXAMPLE 2 Write an equation from a graph STEP 3 Rewrite the equation in standard form. 3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

EXAMPLE 1 GUIDED PRACTICE for Examples 1 and 2 Write two equations in standard form that are equivalent to x – y = 3. 1. SOLUTION To write one equivalent equation,multiply each side by 2. To write another equivalent equation,multiply each side by 3. 2x – 2y = 6 3x – 3y = 9

Complete an equation in standard form
EXAMPLE 3 EXAMPLE 4 EXAMPLE 4 Complete an equation in standard form Find the missing coefficient in the equation of the line shown. Write the completed equation. SOLUTION STEP 1 Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A. Ax + 3y = 2 Write equation. A(–1) + 3(0) = 2 Substitute – 1 for x and 0 for y. –A = 2 Simplify. A = – 2 Divide by – 1.

Complete an equation in standard form
EXAMPLE 4 Complete an equation in standard form STEP 2 Complete the equation. – 2x + 3y = 2 Substitute – 2 for A.

Complete an equation in standard form GUIDED PRACTICE
EXAMPLE 4 EXAMPLE 3 Complete an equation in standard form GUIDED PRACTICE Write an equation of a line for Examples 3 and 4 Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. –4x+By = 7, (–1,1) SOLUTION STEP 1 Find the value of B. Substitute the coordinates of the given point for x and y in the equation. Solve for B. –4x + By = 7 Write equation. –4(–1) + B(1) = 7 Substitute –1 for x and 1 for y. B = 3 Simplify.

EXAMPLE 4 Complete an equation in standard form GUIDED PRACTICE for Examples 3 and 4 STEP 2 Complete the equation. – 4x + 3y = 7 Substitute 3 for B.

EXAMPLE 3 EXAMPLE 4 Complete an equation in standard form GUIDED PRACTICE Write an equation of a line for Examples 3 and 4 Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. Ax+y = –3, (2, 11) SOLUTION STEP 1 Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A. Ax + y = –3 Write equation. A(2) + 11 = –3 Substitute 2 for x and 11 for y. 2A= –14 Simplify. A= –7 Divide each side by 2.

EXAMPLE 4 Complete an equation in standard form GUIDED PRACTICE for Examples 3 and 4 STEP 2 Complete the equation. – 7x +y = –3 Substitute –7 for A.

EXAMPLE 5 Solve a multi-step problem Library Your class is taking a trip to the public library. You can travel in small and large vans. A small van holds 8 people and a large van holds 12 people. Your class could fill 15 small vans and 2 large vans. Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill. a. b. Graph the equation from part (a). c. List several possible combinations.

EXAMPLE 5 Solve a multi-step problem SOLUTION a. Write a verbal model. Then write an equation. 8 s l p 12 + = Because your class could fill 15 small vans and 2 large vans, use (15, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p. 8(15) + 12(2) = p Substitute 15 for s and 2 for l. 144 = p Simplify. Substitute 144 for p in the equation 8s + 12l = p.

EXAMPLE 5 Solve a multi-step problem ANSWER The equation 8s + 12l = 144 models the possible combinations. b. Find the intercepts of the graph. Substitute 0 for s. 8(0) + 12l = 144 l = 12 8s + 12(0) = 144 Substitute 0 for l. 8s + 12(0) = 144 s = 18

EXAMPLE 5 Solve a multi-step problem Plot the points (0, 12) and (18, 0). Connect them with a line segment. For this problem only nonnegative whole-number values of s and l make sense. The graph passes through (0, 12), (6, 8),(12, 4), and (18, 0). So, four possible combinations are 0 small and 12 large, 6 small and 8 large, 12 small and 4 large, 18 small and 0 large. c. 8s + 12(0) = 144

EXAMPLE 5 Solve a multi-step problem GUIDED PRACTICE for Example 5 EXAMPLE 5 Solve a multi-step problem 7. WHAT IF? In Example 5, suppose that 8 students decide not to go on the class trip. Write an equation that models the possible combinations of small and large vans that your class could fill. List several possible combinations.

Solve a multi-step problem GUIDED PRACTICE for Example 5 EXAMPLE 5
SOLUTION STEP 1 Write a verbal model. Then write an equation. 8 s l p 12 + = 8 students decide not to go on the class trip, so the class could fill 14 small vans and 2 large vans. Because your class could fill 14 small vans and 2 large vans, use (14, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p. 8(14) + 12(2) = p Substitute 14 for s and 2 for l. 136 = p Simplify. Substitute 136 for p in the equation 8s + 12l = p.

EXAMPLE 5 Solve a multi-step problem GUIDED PRACTICE for Example 5 ANSWER The equation 8s + 12l = 136 models the possible combinations. STEP 2 Find the intercepts of the graph. Substitute 0 for s. 8(0) + 12l = 136 l = 11 4 12 8s + 12(0) = 144 Substitute 0 for l. 8s + 12(0) = 136 s = 17

GUIDED PRACTICE for Example 5 Plot the point(0, )and(17, 0).connect them with a line segment. For this problem only negative whole-number values of s and l make sense. 4 12 STEP 3 The graph passes through (17, 0), (14, 2), (11, 4), (8, 6), (5, 8) and (2, 10). So, several combinations are 17 small, 0 large; 14 small 2 large; 11 small, 4 large; 18 small, 6 large; 5 small, 8 large; 2 small, 10 large. 8s + 12(0) = 144

Warm-Up – 5.5

Are the lines parallel? Explain. 1. y – 2 = 2x, 2x + y = 7 No; one slope is 2 and the other is –2. ANSWER 2. –x = y + 4, 3x + 3y = 5 ANSWER Yes; both slopes are –1. 2. Graph x + y > 1 ANSWER

3. You play tennis at two clubs. The total cost C (in dollars) to play for time t (in hours) and rent equipment is given by C = 15t + 23 at one club and C = 15t + 17 at the other. What is the difference in total cost after 4 hours of play? ANSWER $6 3. You play tennis at a club. The first two hours are free for members if you pay the $50 yearly fee. After that you pay $10 per hour. Your friend isn’t a member, so she can play for $15 an hour. Write two equations for the cost as a function of time played. Members: C(t)= 10(t-2) + 50 Non-Members: C(t) = 15t ANSWER

6. Find Equation in SLOPE INTERCEPT FORM of the line that goes through (2,8) and (5,17) Y = 3x + 2 ANSWER 2. Write equation in POINT-SLOPE FORM of the line that goes through (2,-6) and (-3, 4) ANSWER Y+6 = -2(x-2) OR y-4 = -2(x+3) 2. Write an equation in STANDARD FORM of the line that passes through (1,3) and (3,13) -5X + Y = -2 ANSWER 89

5.5 - Warmup Take out your graph paper, patty paper, ruler, protractor, and pencil. Graph two points that are on gridlines and connect them. MAKE SURE YOUR SLOPE DOES NOT EQUAL 1. Find the slope of your line. Take one of the two points, and using the protractor make a point that is 90 degrees from your point. Connect these points. Find the slope of your new line. What do we call these types of lines? What do you notice about the slopes of perpendicular lines? But I bet it only works once!!!! *grin*

Vocabulary – 5.5 Conditional Statement
A statement with a hypothesis and a conclusion Frequently posed as an “if-then” statement Example: If Haley is at a volleyball game, then she’s not here. Is this statement always true? Converse A statement that swaps the hypothesis and conclusion of a conditional statement. Example: If Haley’s not here, then she’s at a volleyball game. Perpendicular lines Lines that form a right angle at their intersection.

Notes – 5.5 – Parallel and Perpendicular Lines
If two lines are Parallel, then their slopes are??? Equal What’s the converse?? And is it always true?? If two lines have equal slopes, then they are ??? Parallel OR IDENTICAL!! If two lines are Perpendicular, then their slopes are ? Negative Reciprocals What’s the converse?? Is it always true?? If two non-vertical lines have slopes that are negative reciprocals, they are ??? Perpendicular Yes, it’s always true!

Examples 5.5

EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (–3,–5) and is parallel to the line y = 3x – 1. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (– 3, – 5) has a slope of 3. STEP 2 Find the y-intercept. Use the slope and the given point.

Write an equation of a parallel line
EXAMPLE 1 Write an equation of a parallel line y = mx + b Write slope-intercept form. – 5 = 3(– 3) + b Substitute 3 for m, 23 for x, and 25 for y. 4 = b Solve for b. STEP 3 Write an equation. Use y = mx + b. y = 3x + 4 Substitute 3 for m and 4 for b.

GUIDED PRACTICE for Example 1 1. Write an equation of the line that passes through (–2, 11) and is parallel to the line y = – x + 5. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of – 1.So, the parallel line through (– 2, 11) has a slope of – 1. STEP 2 Find the y-intercept. Use the slope and the given point.

Write an equation. Use y = m x + b.
GUIDED PRACTICE for Example 1 y = mx + b Write slope-intercept form. 11 = (–1 )(– 2) + b Substitute 11 for y, – 1 for m, and – 2 for x. 9 = b Solve for b. STEP 3 Write an equation. Use y = m x + b. y = – x + 9 Substitute – 1 for m and 9 for b.

Determine which lines, if any, are parallel or perpendicular.
EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b: x +5y = 2 Line c: –10y – 2x = 0 SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form. The slope is 5. Write the equations for lines b and c in slope-intercept form.

EXAMPLE 2 Line b: x + 5y = 2 5y = – x + 2 x y = 2 5 1 + – Line c:
Determine whether lines are parallel or perpendicular Line b: x + 5y = 2 5y = – x + 2 x y = 2 5 1 + – Line c: – 10y – 2x = 0 – 10y = 2x y = – x 1 5

Lines b and c have slopes of – , so they are
EXAMPLE 2 Determine whether lines are parallel or perpendicular ANSWER Lines b and c have slopes of – , so they are parallel. Line a has a slope of 5, the negative reciprocal of – , so it is perpendicular to lines b and c. 1 5

GUIDED PRACTICE for Example 2 Determine which lines, if any, are parallel or perpendicular. Line a: 2x + 6y = – 3 Line b: 3x – 8 = y Line c: –1.5y + 4.5x = 6 Find the slopes of the lines. Line a: 2x + 6y = – 3 6y = –2x – 3 x y = 1 2 3 –

GUIDED PRACTICE for Example 2 Line b: 3x – 8 = y Line c: –1.5y + 4.5x = 6 – 1.5y = 4.5x – 6 y = 3x – 4 Lines b and c have slopes of 3, so they are parallel. Line a has a slope of , the negative reciprocal of 3, so it is perpendicular to lines b and c. 1 3 –

EXAMPLE 3 Determine whether lines are perpendicular The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they? STATE FLAG Line a: 12y = – 7x + 42 Line b: 11y = 16x – 52 SOLUTION Find the slopes of the lines. Write the equations in slope-intercept form.

EXAMPLE 3 Determine whether lines are perpendicular Line a: 12y = – 7x + 42 y = – x + 12 42 7 Line b: 11y = 16x – 52 11 52 y = x – 16 ANSWER The slope of line a is – The slope of line b is The two slopes are not negative reciprocals, so lines a and b are not perpendicular. 7 12 16 11

EXAMPLE 4 Write an equation of a perpendicular line Write an equation of the line that passes through (4, – 5) and is perpendicular to the line y = 2x + 3. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 –

Write an equation of a perpendicular line
EXAMPLE 4 Write an equation of a perpendicular line STEP 2 Find the y-intercept. Use the slope and the given point. y = mx + b Write slope-intercept form. – 5 = – (4) + b 1 2 Substitute – for m, 4 for x, and – 5 for y. 1 2 – 3 = b Solve for b. STEP 3 Write an equation. y = m x + b Write slope-intercept form. y = – x – 3 1 2 Substitute – for m and – 3 for b. 1 2

GUIDED PRACTICE for Examples 3 and 4 3. Is line a perpendicular to line b? Justify your answer using slopes Line a: 2y + x = – 12 Line b: 2y = 3x – 8 SOLUTION Find the slopes of the lines. Write the equations in slope-intercept form. Line a: 2y + x = 12 y = – x 1 2 – 6

GUIDED PRACTICE for Examples 3 and 4 Line b: 2y = 3x – 8 y = x 3 2 – 4 ANSWER The slope of line a is – The slope of line b is The two slopes are not negative reciprocals, so lines a and b are not perpendicular. 1 2 3

GUIDED PRACTICE for Examples 3 and 4 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of 4.Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, 3) is 1 4 –

Find the y-intercept. Use the slope and the given point.
GUIDED PRACTICE for Examples 3 and 4 STEP 2 Find the y-intercept. Use the slope and the given point. y = mx + b Write slope-intercept form. 3 = – (4) + b 1 4 Substitute 3 for y, 3 for x, and for y. 1 4 – 4 = b Solve for b. STEP 3 Write an equation. y = m x + b Write slope-intercept form. y = – x + 4 1 4 Substitute – for m and 4 for b. 1 2

Warm-Up – 5.6

Find the slopes of the line that passes through the point. 1. (–4, 1) and (6, –4) ANSWER – 1 2 2. (2, –3) and (–1, 6) ANSWER –3

Find the slopes of the line that passes through the point. 3. Your commission c varies with the number s of pair of shoes you sell. You made $180 when you sold 15 pairs of shoes. Write a direct variation equation that relates c to s. ANSWER c = 12s 3. Graph 3x + 4y <= 12 ANSWER

Vocabulary – 5.6 Scatter Plot
Points that show relationships or trends in data Correlation between data Shows a relationship between data (if it exists) Line of fit AKA Linear Regression Line that “approximates” the data by modeling the trend.

Notes – 5.6 – Fit Data to a Line
Three types of correlation Positive – Trends Up – As x gets larger, y gets larger Negative – Trends Down – As x gets larger, y gets smaller No Trend – Little to no “pattern” or correlation Negative Positive No Trend

Notes – 5.6 – Fit Data to a Line
To construct a regression line, about half the points should be above the line and about half below. The “accuracy” of the line or how close is models the data is given by the r2 value (r is short for residuals.) If the r2 value is VERY close to 1 the regression line accurately models the data. If the r2 value is not close, the data isn’t showing a very strong correlation and the regression line is not very accurate. Once you have the line, you can use points or the graph to determine the equation of the line.

Examples 5.6

EXAMPLE 1 Describe the correlation of data Describe the correlation of the data graphed in the scatter plot. a. a. The scatter plot shows a positive correlation between hours of studying and test scores. This means that as the hours of studying increased, the test scores tended to increase.

EXAMPLE 1 Describe the correlation of data b. b. The scatter plot shows a negative correlation between hours of television watched and test scores. that as the hours of television This means that as the hours of television watched Increased, the test scores tended to decrease.

GUIDED PRACTICE for Example 1 Using the scatter plots in Example 1, predict a reasonable test score for 4.5 hours of studying and 4.5 hours of television watched. 1. Sample answer: 72, 77 ANSWER

EXAMPLE 2 Make a scatter plot Swimming Speeds The table shows the lengths (in centimeters) and swimming speeds (in centimeters per second) of six fish.

EXAMPLE 2 Make a scatter plot a. Make a scatter plot of the data. b. Describe the correlation of the data.

EXAMPLE 2 Make a scatter plot a. Treat the data as ordered pairs. Let x represent the fish length (in centimeters),and let y represent the speed (in centimeters per second). Plot the ordered pairs as points in a coordinate plane. SOLUTION b. The scatter plot shows a positive correlation, which means that longer fish tend to swim faster.

GUIDED PRACTICE for Example 2 Make a scatter plot of the data in the table. Describe the correlation of the data. 2. ANSWER The scatter plot shows a positive correlation.

Write an equation to model data
EXAMPLE 3 Write an equation to model data BIRD POPULATIONS The table shows the number of active red-cockaded woodpecker clusters in a part of the De Soto National Forest in Mississippi. Write an equation that models the number of active clusters as a function of the number of years since 1990. Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 Active clusters 22 24 27 34 40 42 45 51

EXAMPLE 3 Write an equation to model data SOLUTION STEP 1 Make a scatter plot of the data. Let x represent the number of years since Let y represent the number of active clusters.

EXAMPLE 3 Write an equation to model data STEP 2 Decide whether the data can be modeled by a line. Because the scatter plot shows a positive correlation, you can fit a line to the data. STEP 3 Draw a line that appears to fit the points in the scatter plot closely. STEP 4 Write an equation using two points on the line. Use (2, 20) and (8, 42).

EXAMPLE 3 Write an equation to model data Find the slope of the line. m = 11 3 42 – 20 8 – 2 = 22 6 y2 – y1 x2 – x1 Find the y-intercept of the line. Use the point (2, 20). y = mx + b Write slope-intercept form. 20 = (2) + b 11 3 Substitute for m, 2 for x, and 20 for y. 11 3

EXAMPLE 3 Write an equation to model data 38 3 = b Solve for b. An equation of the line of fit is y = 11 3 x + 38 The number y of active woodpecker clusters can be modeled by the y = where x is the number of years since 1990. ANSWER 11 3 x + 38

GUIDED PRACTICE for Example 3 3. Use the data in the table to write an equation that models y as a function of x. ANSWER y = 1.6x + 2.3

EXAMPLE 4 Interpret a model Refer to the model for the number of woodpecker clusters in Example 3. a. Describe the domain and range of the function. b. At about what rate did the number of active woodpecker clusters change during the period 1992–2000?

EXAMPLE 4 Interpret a model SOLUTION a. The domain of the function is the the period from to 2000,or 2 x The range is the the number of active clusters given by the function for 2 x 10, or y < – b. The number of active woodpecker clusters increased at a rate of or about 3.7 woodpecker clusters per year. 11 3

EXAMPLE 4 GUIDED PRACTICE for Example 4 In Guided Practice Exercise 2, at about what rate does y change with respect to x 4. ANSWER y changes with respect to x at the rate of about 1.6

Warm-Up – 5.7

1. Evaluate f(x) = 2.5x + 8 when x is 3 or 5. ANSWER 15.5; 20.5 2. The table shows the profit of a company. Write an modeling the profit y as the function of the number of years x since 1998. ANSWER y = 2.8x

Vocabulary – 5.7 Linear Regression
Line of “best fit” – Approximates the data in plot Interpolation Using a linear regression line to APPROXIMATE a point that is BETWEEN TWO KNOWN VALUES! Extrapolation Using a linear regression line to APPROXIMATE a point that is OUTSIDE of the data range (OR THE KNOWN VALUES). Tells the future!! Zero of a function Where a FUNCTION equals zero.

Notes –5.7 – Predict with Lin. Models
You can use your knowledge of Linear functions and the calculator to predict the future or the past. IMPORTANT CALCULATOR FUNCTIONS: Enter Lists – STATEDIT Plot Lists – 2nd Y= (Stat Plot) Turn Plot on and configure it (SET WINDOW!!) Remember ZOOMSTATPLOT as well. Determine points on a line (e.g. to find y1(x) using function notation or predict the future!) VARSY-VarsFunctionY1 Enter Press ( put value here ) and press Enter Line should look like Y1(5)

Notes –5.7 – Continued IMPORTANT CALCULATOR FUNCTIONS:
To find a Regression Line StatCalcLinreg (ax+b) Type in the lists where you put the data Tell “Calli” where you want to store the eqn. EX: LinReg (ax+b) L1,L2,Y1 Turn Plot on and configure it (SET WINDOW!!) Remember ZOOMSTATPLOT as well.

Examples 5.7 – TURN TO PAGE 335 IN BOOK!

EXAMPLE 1 Interpolate using an equation CD SINGLES The table shows the total number of CD single shipped (in millions) by manufacturers for several years during the period 1993–1997.

EXAMPLE 1 Interpolate using an equation a. Make a scatter plot of the data. b. Find an equation that models the number of CD singles shipped (in millions) as a function of the number of years since 1993. c. Approximate the number of CD singles shipped in 1994.

EXAMPLE 1 Interpolate using an equation SOLUTION a. Enter the data into lists on a graphing calculator. Make a scatter plot, letting the number of years since 1993 be the x-values (0, 2, 3, 4) and the number of CD singles shipped be the y-values. b. Perform linear regression using the paired data. The equation of the best-fitting line is approximately y = 14x

EXAMPLE 1 Interpolate using an equation c. Graph the best-fitting line.Use the trace feature and the arrow keys to find the value of the equation when x = 1. ANSWER About 16 million CD singles were shipped in 1994.

EXAMPLE 2 Extrapolate using an equation CD SINGLES Look back at Example 1. a. Use the equation from Example 1 to approximate the number of CD singles shipped in 1998 and in 2000. b. In 1998 there were actually 56 million CD singles shipped. In 2000 there were actually 34 million CD singles shipped. Describe the accuracy of the extrapolations made in part (a).

EXAMPLE 2 Extrapolate using an equation SOLUTION a. Evaluate the equation of the best-fitting line from Example 1 for x = 5 and x = 7. The model predicts about 72 million CD singles shipped in 1998 and about 100 million CD singles shipped in 2000.

EXAMPLE 2 Extrapolate using an equation b. The differences between the predicted number of CD singles shipped and the actual number of CD singles shipped in 1998 and 2000 are 16 million CDs and 66 million CDs, respectively. The difference in the actual and predicted numbers increased from 1998 to So, the equation of the best-fitting line gives a less accurate prediction for the year that is farther from the given years.

GUIDED PRACTICE for Examples 1 and 2 1. HOUSE SIZE The table shows the median floor area of new single-family houses in the United States during the period 1995–1999. a. Find an equation that models the floor area (in square feet) of a new single-family house as a function of the number of years since 1995. ANSWER y = 26.6x

GUIDED PRACTICE for Examples 1 and 2 b. Predict the median floor area of a new single-family house in 2000 and in 2001. ANSWER about ft2, about 2081 ft2 c. Which of the predictions from part (b) would you expect to be more accurate? Explain your reasoning. ANSWER The prediction for 2000 because the farther removed an x-value is from the known x-values, the less confidence you can have in the accuracy of the predicted y-value.

EXAMPLE 3 Predict using an equation SOFTBALL The table shows the number of participants in U.S. youth softball during the period 1997–2001. Predict the year in which the number of youth softball participants reaches 1.2 million. Year 1997 1998 1999 2000 2001 Participants (millions) 1.44 1.4 1.411 1.37 1.355

EXAMPLE 3 Predict using an equation SOLUTION STEP 1 Perform linear regression. Let x represent the number of years since 1997, and let y represent the number of youth softball participants (in millions). The equation for the best-fitting line is approximately y = – 0.02x

EXAMPLE 3 Predict using an equation STEP 2 Graph the equation of the best-fitting line. Trace the line until the cursor reaches y = 1.2. The corresponding x-value is shown at the bottom of the calculator screen. ANSWER There will be 1.2 million participants about 12 years after 1997, or in 2009.

GUIDED PRACTICE for Example 3 2. SOFTBALL In Example 3, in what year will there be 1.25 million youth softball participants in the U.S? ANSWER There will be 1.25 million youth softball participants in 2006.

Find the zero of a function
EXAMPLE 4 Find the zero of a function SOFTBALL Look back at Example 3. Find the zero of the function. Explain what the zero means in this situation. SOLUTION Substitute 0 for y in the equation of the best-fitting line and solve for x. y = – 0.02x Write the equation. 0 = – 0.02x Substitute 0 for y. x 72 Solve for x.

EXAMPLE 4 Find the zero of a function ANSWER The zero of the function is about 72. The function has a negative slope, which means that the number of youth softball participants is decreasing. According to the model, there will be no youth softball participants 72 years after 1997, or in 2069.

Substitute 0 for y in the equation of the best-fitting line
GUIDED PRACTICE for Example 4 JET BOATS The number y (in thousands) of jet boats purchased in the U.S. can be modeled by the function y = – 1.23x + 14 where x is the number of years since Find the zero of the function. Explain what the zero means in this situation. 3. SOLUTION Substitute 0 for y in the equation of the best-fitting line and solve for x. y = – 1.23x + 14 Write the equation. 0 = – 1.23x + 14 Substitute 0 for y. x Solve for x.

EXAMPLE 4 GUIDED PRACTICE for Example 4 ANSWER The function has a negative slope, which means that the number of jet boats purchased in the U.S. is decreasing. According to the model, there will be no jet boats purchased 11.4 years after 1995, or in 2006.

Review – Ch. 5

Daily Homework Quiz For use after Lesson 5.1 Write an equation of the line with a slope of – 4 and a y-intercept of 1. 1. ANSWER y = – 4x + 1 Write an equation of the line that passes through the given points. (–9,1), (0, –8) 2. ANSWER y = – x –8 (–4, –6), (0, 6) 3. ANSWER y = 3x + 6

Daily Homework Quiz For use after Lesson 5.1 Write an equation for the linear function f with f (0) = – 4 and f(–1) = –9. 4. ANSWER y = 5x – 4 An electronics game store sells used games for $12.99 with a $20 membership fee. Write an equation that gives the total cost to become a member and buy games as a function of the number of games that are purchased. Then find the cost or 6 games. 5. ANSWER C =12.99g + 20 where C is total cost and g is the number of games; $97.94

Daily Homework Quiz For use after Lesson 5.2 Write an equation of the line that passes through the given point with given slope. (4, –1), m = –1 1. ANSWER y = – x + 3 (2, 0), m = 4 2. ANSWER y = 4x – 8

Daily Homework Quiz For use after Lesson 5.2 Write an equation of the line that passes through the given point. (2, 3), (4,7) 3. ANSWER y = 2x – 1 (–5, 7), (2, –7) 4. ANSWER y = –2x – 3 A camp charges a registration fee and a daily amount. If the total bill for one camper was $338 for 12 days and the total bill for another camper was $506 for 19 days and the total bill be for a camper who enrolls for 30 days? 5. ANSWER $770

Daily Homework Quiz For use after Lesson 5.2 Write an equation in point-slope form of the line that passes through (6, – 4) and has slope 2. 1. ANSWER y + 4 = –2(x –6) Write an equation in point-slope form of the line that passes through (–1, –6) and (3,10). 2. ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

Daily Homework Quiz For use after Lesson 5.2 A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip. 3. ANSWER C = 235t + 170, where C is total cost and t is time (in days); $1815

Daily Homework Quiz For use after Lesson 5.4 Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points. 1. (1, – 6), m = – 2 ANSWER 2x + y = – 4 2. (– 4, – 3), (2, 9) ANSWER – 2x + y = 5

Daily Homework Quiz For use after Lesson 5.4 3. You have $96 to spend on campground activities. You can rent a paddleboat for $8 per hour and a kayak for $6 per hour. Write an equation in standard form that models the possible hourly combinations of activities you can afford. List three possible combinations. ANSWER 8p + 6c = 96; 16 h kayak and 0 h paddleboat; 12 h paddleboat and 0 h kayak; 6 h paddleboat and 8 h kayak.

Daily Homework Quiz For use after Lesson 5.5 1. Write an equation of the line that passes through the point (–1,4) and is parallel to the line y = 5x –2. y = 5x + 9 ANSWER Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x +2. 1 4 – 2. y = 4x + 3 ANSWER

Daily Homework Quiz For use after Lesson 5.5 3. Path a, b and c are shown in the co ordinate grid. Determine which paths,if any, are parallel or perpendicular. Justify your answer using slopes. ANSWER Paths a and b are perpendicular because their slopes, and 2 are negative reciprocals. No paths are parallel. 1 2 –

Daily Homework Quiz For use after Lesson 5.6 1. Tell whether x and y show a positive correlation, a negative correlation, or relatively no correlation. ANSWER negative correlation.

Daily Homework Quiz For use after Lesson 5.6 2. The table shows the body length and wingspan (both in inches) of seven birds. Write an equation that models the wingspan as a function of body length. y = 3.1x – 10.3, where x is body length and y is wingspan. ANSWER

Warm-Up – X.X

Vocabulary – X.X Holder Holder 2 Holder 3 Holder 4

Notes – X.X – LESSON TITLE.
Holder

Examples X.X

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